Transcript Trig Identites with Multiply Conjugate

The Fundamental Identities
Fundamental Trigonometric Identities
Reciprocal Identities
1
csc x
1
csc x 
sin x
sin x 
1
sec x
1
sec x 
cos x
1
cot x
1
cot x 
tan x
sin x
cos x
cos x
sin x
cos x 
tan x 
Quotient Identities
tan x 
cot x 
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The Fundamental Identities cont’
Fundamental Trigonometric Identities
Pythagorean Identities
sin2 x + cos2 x  1
1 + tan2 x  sec2 x
1 + cot2 x  csc2 x
*The identities in the above slides are given on the formula sheet*
Example: Changing to Sines and Cosines to Verify an Identity
Verify the identity: sec x cot x  csc x.
Solution
The left side of the equation contains the more complicated
expression. Thus, we work with the left side. Let us express this side of the
identity in terms of sines and cosines.
sec x cot x 
1
cos x

cos x sin x
Apply a reciprocal identity: sec x = ¹/cos x
and a quotient identity: cot x = cos x/sin x.
1

1
cos x

cos x sin x
Divide both the numerator and the
denominator by cos x, the common factor.
1
1
sin x
 csc x

Multiply the remaining factors in the
numerator and denominator.
Apply a reciprocal identity: csc x = ¹/sin x.
By working with the left side and simplifying it so that it is identical to the
right side, we have verified the given identity.
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Example: Using Factoring to Verify an Identity
Verify the identity: cos x - cos x sin2x  cos3 x.
Solution
We start with the more complicated side, the left side. Factor out
the greatest common factor, cos x, from each of the two terms.
cos x - cos x sin2 x  cos x(1 - sin2 x)
 cos x ∙
 cos3 x
cos2
x
Factor cos x from the two terms.
Use a variation of sin2 x + cos2 x = 1.
Solving for cos2 x, we obtain cos2 x =
1 – sin2 x.
Multiply.
We worked with the left and arrived at the right side. Thus, the identity is
verified.
Example
• Verify the identity:
cos x  cos x + cos x sin x
3
2
Solution:
cos x  cos x + cos x sin x
3
2
cos x  cos x(cos x + sin x)
2
cos x  cos x(1)
2
Example: Multiplying the Numerator and
Denominator to Verify an Identity
Verify the identity:
sin x
1 - cos x

.
1 + cos x
sin x
Solution
The suggestions given in the previous examples do not apply
here. Everything is already expressed in terms of sines and cosines.
Furthermore, there are no fractions to combine and neither side looks more
complicated than the other.
Let's solve the puzzle by working with the left side and making it look like the
expression on the right. The expression on the right contains 1 - cos x in the
numerator. This suggests multiplying the numerator and denominator of the left
side by 1 - cos x. By doing this, we obtain 1 - cos x in the numerator, the same
as the numerator on the right.
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Example: Multiplying the Numerator and
Denominator to Verify an Identity
Verify the identity:
sin x
1 - cos x

.
1 + cos x
sin x
Solution
sin x
sin x
1 - cos x


1 + cos x 1 + cos x 1 - cos x
sin x(1 - cos x)

1 - cos 2 x
sin x(1 - cos x)

sin 2 x
1 - cos x

sin x
Multiply the numerator and denominator by 1 – cos x.
Multiply. Use (A + B)(A – B) = A2 – B2, with A = 1 and
B = cos x, to multiply denominators.
Use a variation of sin2 x + cos2 x = 1. Solving for
sin2 x, we obtain sin2 x = 1 – cos2 x.
Simplify:
sin x
sin 2 x
=
sin x
sin x  sin x
=
1
sin x
We worked with the left and arrived at the right side. Thus, the identity is
verified.
Guidelines for Verifying Trigonometric Identities
1. Work with each side of the equation independently of the other side. Start
with the more complicated side and transform it in a step-by-step fashion
until it looks exactly like the other side.
2. Analyze the identity and look for opportunities to apply the fundamental
identities. Rewriting the more complicated side of the equation in terms of
sines and cosines is often helpful.
3. If sums or differences of fractions appear on one side, use the least common
denominator and combine the fractions.
4. Don't be afraid to stop and start over again if you are not getting anywhere.
Creative puzzle solvers know that strategies leading to dead ends often
provide good problem-solving ideas.